Competitive companies understand that
satisfying customers' needs and running a successful operation requires a
system that is dependable, predictable and consistent. Many of todays
quality management philosophies, such as Six Sigma, focus on reducing
variations and scrap. A modern view of quality defines it as the inverse of
variability. Therefore, measuring variability and finding ways to reduce it
is important for the success of a company. This article presents a Weibull
analysis approach to modeling production data for a process in order to
assess its deviation and capabilities. This approach augments the Six
Sigma-type calculations, which typically rely on the normality assumption.

We have discussed in past issues how
process availability and throughput can be studied through the modeling of
the process (for example, see
Issue 28s
Reliability Basics article). This "in-depth" way of assessing a process
yield can be complemented by a "higher view" approach that relies on using
actual periodical production data of the system. This is a quick and useful
way to assess the "health" of a process and it is a stepping stone for
getting the process as close to the target as possible.

Introduction
Where does the "Sigma" in Six Sigma come from? Sigma is a statistical
notation for the standard deviation parameter that can be used to quantify
how far a given process deviates from perfection. The central idea behind
Six Sigma is measuring the number of "defects" in a process and figuring out
how to eliminate them in order to get the process more in line with its
goal. "Defects" can be used in a more general context; the term could refer
to any issues that cause concern to a company, such as defective products,
out-of-specs products, inconsistency in production volumes, etc. Measuring
defects also leads to discussions about the capability of the process to
avoid defects, i.e. how far the process is from being the on-target,
predictable and controlled process that the company hopes to have.

The next figure compares production volumes
of two processes, a healthy one and a troubled one.

Figure 1 Two Different
Processes

In Six Sigma, variations are
typically modeled with the bell-curve normal distribution. Oftentimes,
processes and products are not "well-behaved" and their distributions are
typically skewed. The skewness is generally more apparent when the
variations are significant. Therefore, Six Sigma calculations have been
criticized for relying heavily on the normality assumption.

In this article, we will focus
on the Weibull distribution as our distribution model because of its
flexibility and generally skewed pdf shape, as shown in Figure 2;
however, other types of distributions (such as the lognormal, Gumbel and
logistic distributions) might be appropriate depending on the types of
variation.

Figure 2 Normal and
Weibull pdf Plots Comparison

The following items are some
of the key characteristics to look at when studying the probability plots of
the process. These characteristics can help you discover signs of an
unhealthy process.

Deviation from the
target:The next plot provides a good way to detect shift compared to an
acceptable process that is centered. It also gives a visual idea of the
production loss, inefficiencies and lost potential of the process.

Figure 3 Process Showing
Significant Production Losses

Steepness of probability
line or width of pdf: This indicates the level of variation in the production. β is
the slope parameter of the Weibull probability line. A small β
value means a flatter probability line and wider pdf plot, all of
which are signs of large variations and an unhealthy process. βs
sister in the normal distribution is σ. In a process that follows
a normal distribution, the smaller the σ, the better.

Figure 4 Probability Plot
for Processes with Different Variations Level

Figure 5 pdf Plot
for Processes with Different Variations Level

Signs of more causes of
variation:Look for curvature in the probability plot. This indicates that the
production is not consistent in its variation and that there are
different factors at work.

Figure 6 Process Showing
Mixed Variations Behavior

Such processes can be
modeled with a mixed distribution such as the mixed Weibull
distribution, which fits a set of Weibull distributions to
subpopulations within the entire population and finds the percentage of
each subpopulation. The same process shown in Figure 5 is modeled with a
mixed Weibull distribution with two subpopulations; the probability plot
is shown next.

Example
A chemical manufacturer collected a sample from a years worth of production
output logs. The manufacturers customer requires that the daily production
volume should be at least 975 tons but no more than 1025 tons, in order to
avoid unnecessary storage costs and deterioration problems. The data set is
modeled in Weibull++.

Figure 8 Process Sample
Production Data

The above graph shows that the
process is off-target. It also does not follow the normal distribution, but
rather a Weibull distribution with β=26.01. This indicates that the
process suffers from significant variations.

Metrics such as Cp,
Cpk and Cpm are common Six Sigma metrics used to assess the
capabilities of a process. A great deal of importance is often assigned to
these metrics, but they can be flawed because they assume that a process
follows the normal distribution. Although some remedies have been suggested,
such as using data transformation methods or modifying the way the process
capability measures are calculated [1], these remedies remain unrobust and
may complicate the analysis. We suggest using probability calculations, as
they are straightforward and do not violate the assumptions of the chosen
distribution. For this example, the percentage of daily productions that met
the goal of 975 tons to 1025 tons can be found using the following
probability calculations:

P(975<p<1025) = P(p<1025)
P(p<975)

where p is a random
variable that describes daily production. Calculating P(p<a) in
Weibull++ means calculating the probability of failure by a.
P(p<1025) and P(p<975) are found using Weibull++s QCP as follows.

Therefore, P(975<p<1025) =
P(p<1025) P(p<975) = 0.998654 0.834599 = 0.164055 or 16.4055%. In other
words, 83.5945% of the days in the sample did not meet the production
requirements.

It is important to convert
your statistical analysis into money figures, which will have more impact on
management and are more likely to motivate actions. Think about defects
cost, scrapping cost, rework cost, penalties, lost revenue, customer
dissatisfaction, etc. For this example, the manufacturer is penalized by its
customer $1000 for every day of insufficient production and $500 for every
day of excessive production that requires storage. For a one year period,
the penalties amount to:

Penalties = 365.[P(p<975).$1000
+ P(p>1025).$500]

Which is equivalent to:

Penalties = 365.[P(p<975).$1000
+ (1-P(p<1025)).$500]

Therefore:

Penalties = 365.[0.834599.$1000
+ (1-0.998654).$500]= $304,874.3

Conclusion
This article presented a Weibull analysis approach to model production data
for a process. This is used to assess the variability in the process and
discover a number of other types of signs of an "unhealthy" process. This
approach does not rely on the questionable normality assumption in many Six
Sigma calculations and provides an alternative to process capabilities
calculations.